Introduction to Calculus on Manifolds   MATH  295E

Instructor: Ilya Kapovich

Spring 2004  Section E

Textbook:   Micahel Spivak, Calculus on Manifolds

Office:  Illini Hall 328

E-mail: kapovich@math.uiuc.edu

Tel. 265-0633

Office hours:  Monday, Wednesday 4-5:30pm (and at other times by appointment)

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APPROXIMATE    SYLLABUS
 

Prerequisites:
I. Math 295A,B, and C
    or else
II. Math 315, 347, and one of Math 243 or 280.
 
 
 

• Chapters 1-3 (12 hours)

Functions on Euclidean spaces, continuity, differentiation, inverse and
implicit function theorems, integration, Fubini's theorem, change of
variable formula, partition of unity. Proof of the Implicit Function
Theorem. Not all proofs need be covered.

Optional extra material: Sard's theorem.
 

• Ch. 4.1 (3 hours)

Linear algebra: multilinear functions, tensor products, inner products,
exterior algebras and exterior products.
 

• Ch. 4.2 (4 hours)

Vector fields and differential forms on Euclidean space. Closed and
exact forms.
Extra material: Explain how Maxwell's Laws can be written in terms of
differential forms.
 

• Ch. 4.3-4.4 (4 hours)

Standard and singular cubes, singular chains. Stokes' theorem for
integrals over singular chains.
 

• Ch. 5.1-5.2 (8 hours)

Manifolds (as subsets of the Euclidean space admitting atlases with
particular properties of transition functions). Tangent spaces, vector
fields and differential forms on manifolds. Orientation.
Extra material: Matrix groups as submanifolds.
 

• Ch. 5.3-5.4 (8 hours)

Integration on manifolds. Stokes' theorem. Volume form. Classical
theorems (Green, divergence, integration by parts, etc) as corollaries
of Stokes' theorem.

Extra material: Some applications to physics. Gauss's law: Flux
of an electric field through a closed surface is proportional to the
electric charge enclosed by the surface. Faraday's law (about the change
of electric flux through a moving surface with boundary).
 
 

• Exams (4 hours)

Total: 43 hours.